Parametric Equations and Motion

Title of lesson: PARAMETRIC EQUATIONS AND MOTION

Author: Theresa Hogan

Length of lesson: 4 days

Grade Level: advanced Algebra II / Precalculus

Concepts:Parametric equations may be used to describe situations in which a third variable is introduced. By building tables, and plotting points, one can see the relationships between parametric equations and rectangular equations.

Objectives:

The student will be able to:

• Graph parametric equations by hand.
• Transform parametric equations into rectangular equations.

Sources for all four days:

• It’s Going, Going, http://www.pbs.org/teachersource/mathline/concepts/basesoftball/activity3.shtm
• Sullivan, Michael. Pre-Calculus: Graphing and Data Analysis Student Additions. © 1998 Prentice Hall (teacher resource) pgs 726-742.
• Texas Essential Knowledge and Skills

http://www.tenet.edu/teks/math/teks/teksprecal.html

TEKS: (these are Precalculus TEKS, but the lesson does not include trigonometry)

(5)C. convert between parametric and rectangular forms of functions and

equations to graph them;

Materials List and Advanced Preparations:

For the class:

- pre-test

- poster board and markers

- animal cut outs (cow, pig, sheet, cat, duck, turkey, bee, and seal)

- warm-up worksheet (corresponds to exploration activity)

- homework worksheet (problems where students graph by hand parametric equations, and eliminate the parameter of an equation.)

Engagement:

Pre Test will be given at the beginning of class.

Exploration:

What the Teacher Will Do	What the Students Will Do	Ongoing Evaluation (Questions to ask students)
Teacher will randomly distribute animals to the students as they enter the classroom, and that animal will be the group that they are with for that day.	Students will get into groups according to their animal.
Teacher will pass out poster board, markers, and worksheets. (“WORKSHEET 1”) Teacher will also have a grab bag of equations for the students to pull from. (“ACTIVITY 1”) In their groups, the students will graph their parametric equation and rectangular equation by hand, by putting values into a chart and graphing by hand the results. Teacher will walk around classroom and answer any questions about the assignment.	Students will work on their equation as a group.
Teacher will have students present their poster board to the class.	Students will present their poster board with the class. They will explain how they arrived at graphs from the two equations, and any reactions they had while working the problem. Other students will be actively listening to speaker and filling in their worksheet with the information being presented.	What did you discover about the two graphs? The graphs look the same, but why do their equations look so different?

Explanation:

Students present answers and their reasoning behind how they graphed both equations. Students will discover the relationship between the two equations, and how to convert from a parametric equation to a rectangular equation. (Students will see that eliminating the parameter involves writing a single equation in terms of x and y and removing reference to t.)

Each group will take their original equations and present the algebra needed to get from one equation to the next. One student from each group will present their finings.

Elaboration:

Have students discover how to convert the other way, from a rectangular equation to a parametric equation.

Evaluation:

Teacher will monitor student progress throughout the activity, and will evaluate their understanding by having them present, and correct their answers. Teacher will assign homework problems of choice from book (that require students to draw graphs), and determine whether or not the students understand the concepts.

Day 2 and 3

Concepts: Parametric equations can be graphed in a calculator to give students a better understanding of how parametric graphing works. They will be able to compare their hand drawn graphs with the results of the calculator.

Objectives:

The student will be able to:

• Graph parametric equations on their calculator.
• Compare and contrast the windows in parametric mode, from the function mode.
• Discuss the direction in which the parametric graph is moving, why it is important to know, and how it can be altered from changing viewing values.

TEKS:

(5)D. Students use parametric functions to simulate problems involving motion.

Materials List and Advanced Preparations:

For the class:

- Overhead calculator

- Candy

- Overhead transparency

Engagement:

Warm up activity: students will represent a function using a set of parametric equations (i.e. y=x³+1). “Is the relation a function? Is it one-to-one?”

Exploration:

What the Teacher Will Do	What the Students Will Do	Ongoing Evaluation (Questions to ask students)
While students work on the warm up, teacher will walk around class and check homework for completion.	Students will work on warm up.
Teacher will discuss how graphing parametrics by hand is okay in simpler situations, and ask the students what they should do when values become too tedious to plug in.	Students will respond by saying you can use your calculator.	Teacher will ask, can we put this into “y=”? Why not? (students will respond that the parametric will not fit because of “x=”)
Teacher will formally introduce the word ‘parametric,’ and the mode in the calculator that will help us graph these equations.	Student will actively listen, and follow along with their calculator.	How is graphing of parametric equations different from the graphing of a function in two variables?
Teacher will have students refer to their homework from the night before, and using the calculator, will check together to make sure their hand-drawn graph matches up with their calculators graph.	Students will check their homework, and ask for clarification if one graph doesn’t look like the other.	Teacher will ask if students are able to view their graphs on the calculator. Teacher will have students compare and contrast the windows in parametric mode with what they are familiar with in the function mode.
On the last homework problem that they check as a group, Teacher will ask “What if this graph represented the motion of a particle? What can you tell me about its behavior?”	Students should discover that they can trace on the graph to figure out where the particle is increasing the fastest, also they can determine the direction in which is the particle is moving.	In a situation like this, what does t represent? Change the t step to 0.1 instead of 1. What does this do to your graph?
Teacher will give the class practice problems in order for them to play around with the new windows, and get comfortable with it.	Students will actively graph the equations, and give their suggestions as to what a good window would be.	On the last problem, go into detail about the t step again. What would the t step look like if it was 1? What if it was .1? Why does one look smooth and the other look jagged? Manipulate the t step again. Which one is faster to graph? Why is it faster?

Explanation:

· Students will compare and contrast the windows used in the function mode to those of parametric mode.

· Students will explain the relationship of the t step to its parametric graph.

Elaboration:

Students will participate in an activity where they will discover that all of three parametric equations match the same graph. (Teacher picks 3 equations, and splits the class into 3 sections, asking each section which equation they think matches the graph (one equation per section) and whichever section is right gets candy. Of course everyone will say “our equation is right!” so everyone gets candy) Teacher will then explain how parametric equations defining a curve are not unique, and that different parametric equations could represent the same graph. Teacher will ask if this is the same with rectangular equations.

Evaluation:

Teacher will give homework (“HOMEWORK 1”) with problems that require students to play around with values of t steps, and do rough sketches of the graph including the direction of motion, starting and finish points. Students will use sine and cosine in the homework, but will not need to understand anything more than how to type them in on the calculator. Also, students will then try to write an equation to try and change where the equation starts, and even change its direction. This will lead into discussion on the fourth day.

Day 4

Concepts: Parametric equations can be used to model real life situations. They can be used to describe the path of an object relative to time.

Objectives:

The student will be able to:

• Apply prior knowledge about parametrics to a real life situation.
• Students will connect concepts of the Pythagorean Theorem and physics to solve the problem.
• Students will be able to graph their equations in the calculator, and think critically of how the movement of an object would look graphically.

TEKS:

(5)D. use parametric functions to simulate problems involving motion.

Materials List and Advanced Preparations:

For the class:

- baseball paper cut-outs

- baseball in-class worksheet

- post-test (“POST-TEST”)

Engagement:

Go over the previous night’s homework, see what equations they got, answer any questions, and collect all homework. Teacher will engage class by introducing the baseball problem, go over projectile motion and how it is represented in physics and in later mathematics. Teacher will shortly explain the concept of sine and cosine (and review one-to-one functions) realizing the students have not seen sine/cosine yet, and that “theta” is just another parameter, like t or any other letter. The equations for projectile motion will be on the worksheet (see attached).

Exploration:

What the Teacher Will Do	What the Students Will Do	Ongoing Evaluation (Questions to ask students)
When the students entered the class they picked up a baseball with the team written on it, and will be divided into groups based on which team they got. Teacher will stress that only one worksheet from each group will be taken up, and that worksheet will be randomly chosen. (This will help to make sure that all students are working.)	Students will get into groups and begin the assignment. Students’ task is to find out if the baseball will clear the fence, so the students will be actively working in groups to determine whether or not it did.
Teacher will walk around classroom, and monitor students’ progress.		Students may not know how to make this into a parametric equation. Teacher will stress to individual groups that you must break down the problem into horizontal and vertical components. Teacher will hint for students to use cosine, sine, and velocity if necessary.

Explanation:

The first group to finish will be asked to volunteer their answer on the board, and their reasoning to why they chose that specific equation. Teacher will ask other groups if they agree, or disagree, and walk them through the problem to make sure everyone knows how to solve such a problem.

Elaboration:

Teacher will ask students to determine how long it took for the ball to reach the wall, by how much the ball cleared the fence.

Evaluation:

Teacher will monitor student’s progress and participation throughout the activity, and will evaluate their understanding by having them present their ideas.

WORKSHEET 1

x=

y=

t

x

y

y(x)=

or

x(y)=

ACTIVITY 1

t	x	y
0
1
2
3
4
5
6
7
8

x = 3t + 2 ; y = ⅓ (x + 1)

y = t + 1 2 ≤ x ≤ 26

t	x	y
0
1
2
3
4
5
6
7
8

x = t - 3 ; y = 2x + 10

y = 2t + 4 -3 ≤ x ≤ 5

t	x	y
0
1
2
3
4
5
6
7
8

x = t + 2 ; y = 2 ≤ x ≤ 10

y =

t	x	y
0
1
2
3
4
5
6
7
8

x = ; y = 2x²

y = 4t 0 ≤ x ≤ 4

t	x	y
-4
-3
-2
-1
0
1
2
3
4

x = t² + 4 ; y = x - 8

y = t² - 4 0 ≤ x ≤ 20

t	x	y
0
1
2
3
4
5
6
7
8
9

x =+ 4 ; y = x - 8

y = - 4 0 ≤ x ≤ 7

t	x	y
-4
-3
-2
-1
0
1
2
3
4

x =3t² ; nope

y = t + 1

t	x	y
-4
-3
-2
-1
0
1
2
3
4

x = 2t - 4 ; y = (x + 4)²

y = 4t² -12 ≤ x ≤ 4

HOMEWORK 1

Time Flies!!!

You get to play with your calculator for this assignment J The purpose of this activity is to manipulate the sine and cosine functions in parametric equations. The goal of this activity is to find a set of parametric equations and settings that resemble the hands of a clock- tracing a clockwise path beginning at the “12.” But let’s play around first…

Do this with your calculator:	Describe the graph(s) you get:
1. Make sure you are in parametric mode, and we are going to be in degrees, not radians. For x(t), type cos(t), and for y(t), type sin(t). In the window settings, let tmin = 0, tmax = 360, and a tstep of 10 is fine. A good window for these problems is -2x2, and -1y1. Graph.	(example of a good description) The graph is a circle of radius = 1, beginning on the positive x-axis (0 degrees), tracing a counter-clockwise path and ending in the same spot.
2. Make cosine negative first, and then make it positive again, with sine negative. Describe both graphs, thinking about how they are different from each other, and how each of them differ from problem 1. How does making one negative seem to affect the graph?
3. Go back to the equations for problem 1, x(t) = cos(t) and y(t) = sin(t). This time, you’re going to change the window settings. Set tmin to -90, and tmax to 270. Now graph it, and describe what you see. How does it differ from the graph of problem 1? (regraph if you need to) What did subtracting 90 from each t value do?
4. Make another change to the equations and/or window, and write down what they are. Equations: Window values: Describe the graph.
5. Based on your observations, how does a negative affect the graph? What if both values were negative? What does switching sine and cosine do? Describe how different ranges of t affect the graph. What does t represent here?

Now for the goal….

Use your new knowledge of modifying different aspects of the graphs, and find a set of parametric equations to describe the movement of the hand of a clock starting at the “12.” Be sure to include all your window and t values.

Could there be more than one set of equations and window values to describe this situation? Can you think of an explanation why? If your answer is yes, try to find another set to support your answer.

Student Handout for “The Baseball Lesson”

Bottom of the ninth- 2 outs, a runner on second base, and the team is down 3 to 2. The center field wall is 400 feet away and 10 feet tall. The batter looks like he is going to swing for a home run! The batter is up at the plate… here comes the pitch… he swings and hits the ball 3 feet above the plate at a solid 35 degree angle with a velocity of 118.25 feet per second straight away to center field!

Does he win his team the game or does he make the last out?

Hint: you’re going to need to use the equations for projectile motion:

x= (v₀cosθ)t y= -½gt² + (v₀sinθ)t + h

POST-TEST

To Sum it All Up….

Do problems on a separate sheet of paper (clearly there is not enough room here) and draw pictures!!!!

1. The height of a falling object is given by the equation y = -16t² + h₀ where t is the time (measured in seconds) and h₀ is the initial height of the object (measured in feet). Find the height of an object dropped from the indicated height after the indicated amount of time.

a. h₀ = 65, t = 1.5

b. h₀ = 100, t = 2.4

c. h₀ = 426, t = 4.7

2. Find all positive solutions for each equation. Round to the nearest hundredth.

a. -16t² + 11.25 = 0

b. -4.9t² + 9.8t + 30 = 0

3. Write parametric equations to simulate the following motion: A golfer swings a club at an elevation of 42° and an initial velocity of 128 ft/sec on level ground.

4. Hayden rolls a ball off the edge of the roof of a 75 ft tall building at an initial velocity of 6.5 ft/sec.

a. Write parametric equations to simulate this motion.

b. What equation can you solve to determine when the ball hits the ground? (hint: when y=0)

c. How long after it rolls off the roof does the ball hit the ground? (Round to the nearest hundredth.)

d. How far from the base of the building does the ball hit the ground? (assume the edge of the building is x=0) (Round to the nearest hundredth)